Definition df-termc 50399

Description: Definition of the proper class (termcnex 50502) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 50412). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50441). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50446). See also dftermc3 50457 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50457, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 50417.

Followed directly from the definition, terminal categories are thin (termcthin 50403). The opposite category of a terminal category is "almost" itself (oppctermco 50431). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50466).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50427) and terminal objects (dftermo4 50428). The universal properties provide an alternate proof of initoeu1 18100, termoeu1 18107, initoeu2 18105, and termoeu2 50164. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50442).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17776, 0thincg 50384, func0g 50015, 0funcg 50011, and initc 50017.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 50398	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1569	. . . . . 6 class 𝑐
4		cbs 17301	. . . . . 6 class Base
5	3, 4	cfv 6533	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1569	. . . . . 6 class 𝑥
8	7	csn 4584	. . . . 5 class {𝑥}
9	5, 8	wceq 1570	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1812	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 50343	. . 3 class ThinCat
12	10, 2, 11	crab 3412	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1570	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is used by:  istermc  50400